At secondary eclipse the removed monochromatic planetary flux is $\pi B_\lambda(T_p)R_p^2/d^2$, while the stellar flux is $\pi B_\lambda(T_s)R_s^2/d^2$. The thermal <exoplanet secondary eclipse> depth is therefore
$$
\boxed{\delta_{\rm ecl}(\lambda)
=\left(\frac{R_p}{R_s}\right)^2
\frac{B_\lambda(T_p)}{B_\lambda(T_s)}}.
$$
A hot Jupiter has tiny visible thermal contrast, rapidly improving contrast toward the infrared, and molecular absorption or emission features superposed on its continuum. Reflected light may add a visible component. In the <Rayleigh-Jeans law> limit, $B_\lambda(T)\propto T/\lambda^4$, so
$$
\boxed{\delta_{\rm ecl}\longrightarrow
\left(\frac{R_p}{R_s}\right)^2\frac{T_p}{T_s}},
$$
a wavelength-independent asymptote for ideal blackbodies.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-315/1/b/solution